484 M. H. Weber on the Heat-conducting Power 



in which (for shortness) we have put 



Zo= : r Zi= z— 



e * — e k — e'^ k~e/ k 



If in the expression for Vg^ we suppose = T, while for Vg^+i 

 it is supposed =0, the two expressions will be equal to one an- 

 other. It is the same when = T in the expression for V2/A+1 

 while ^ = in that for Vg^; and hence, in fact, the final state 

 of the bar in one period is the initial state of the following*. 



It follows from the above expressions that the temperature in 

 the middle of the bar remains constant during each period, and 

 preserves the same value in the even and the odd periods. That 



is to say, for^= - we obtain from the equations (2), for each 



of the two periods, if Yj^ denotes the temperature of the middle 

 of the bar : — 



v.=u+(^~u) 



2 



e^^^ k -{-e ^^ k 



By observing V^, and U, Uq, and u^ being given, the ratio - 



K 



can very easily be deduced ; for if we put, for shortness, 

 2 



m 



then 



k~l'l io^^ J • • • • (3) 



In this we have one equation between the quantities h and k ; 

 it remains to construct a second between the same. Such an 

 equation can be derived in various ways from the equations (2). 

 On closer consideration, however, we find that it is far the most 



* We may easily satisfy ourselves of this by employing the series 



I' Vn I 



which is convergent for all values of x between and I. 



